The invention is directed to portfolio optimization and techniques associated with optimization processing of asset portfolios, such as portfolios of securities, for example.
Markowitz's portfolio theory, a foundation of modern finance, is based on a trade-off between a single return and a single risk measure. Variance or standard deviation of return is typically employed as a measure of risk. The goal of portfolio optimization is maximizing return and, at the same time, minimizing risk. This is typically treated as a two-objective optimization problem using the following problem formulation:
MinimizeVariance;subject toReturn ≧ target, and portfolio constraints.
Alternatively, the portfolio optimization might be described with the following alternative portfolio optimization problem formulation:
MaximizeReturn;subject toVariance ≦ target, and portfolio constraints.
As is known in the art, the efficient frontier can be obtained by varying the return target, or alternatively the risk target, and optimizing on the other measure, i.e., the measure that is not being varied. The resulting efficient frontier is a curve in a two-dimensional space as shown in FIG. 1. Each of these approaches uses variance as a sole risk measure. Each point on the efficient frontier is a portfolio consisting of a collection of assets. For example, these assets may be a collection of securities.
However, there are major drawbacks to this approach. In modern portfolio management, portfolio managers not only care about the variation around mean, but also the risk of losing most of the portfolio's value due to rare events. In a normal situation, the portfolio value fluctuates around its mean due to market volatility and other risk drivers. However, a portfolio may lose a significant amount of value from a low-probability-high-impact event. This possibility calls for a need to use other risk measures, in addition to variance, for managing the portfolio risk.
FIG. 28 shows further aspects of the Pareto optimal front. Most real-world optimization problems have several, often conflicting, objectives. Therefore, the optimum for a multiobjective problem is typically not a single solution; rather, it is a set of solutions that trade off between objectives. This concept was first formulated by the Italian economist Vilfredo Pareto in 1896 (Tarascio, 1968), and it bears his name today. A solution is Pareto optimal if (for a maximization problem) no increase in any criterion can be made without a simultaneous decrease in any other criterion (Winston, 1994). The set of all Pareto optimal points is known as the Pareto optimal front.
FIG. 28 represents the objective space for two dimensions of an imaginary portfolio, where one wants to maximize both yield measure one and yield measure two, for example. The shaded area represents the feasible region (the region where solutions are possible). Solutions A and B are Pareto optimal: no increase in yield measure one can be made without a decrease in yield measure two, and vice versa. Solution C is dominated (not Pareto optimal): there are solutions with the same yield measure two with a higher yield measure one (e.g., solution B), or with a higher yield measure two for the same yield measure one (e.g., solution A). The heavy line 2802 indicates the Pareto optimal front—each point on it is non-dominated. Given the Pareto optimal front, a portfolio manager can choose a solution based on other criteria (e.g., cost of implementing the portfolio, risk measures, and other return measures).
In accordance with further known aspects of portfolio processing, FIG. 33 is a diagram showing aspects of different spaces as is known in the art. FIG. 33 shows an example of linear convex space, nonlinear convex space, as well as nonlinear nonconvex space. Further, FIG. 33 provides a description of such defined spaces, as well as illustrative equations representing such defined spaces. Further, FIG. 33 shows further aspects of each space, i.e., in terms of variables and approaches to solving problems in each particular space.
In a similar manner, in accordance with further known aspects of portfolio processing, FIG. 34 is a diagram showing aspects of different functions as is known in the art. FIG. 34 shows an example of linear function, nonlinear convex function, as well as nonlinear nonconvex function. Further, FIG. 34 provides a description of such functions, as well as illustrative equations representing such defined spaces. Further, FIG. 34 shows further aspects of each space, i.e., in terms of variables and approaches to solving problems in each particular space.
In prior art techniques, various risk measures were developed for capturing tail risk, i.e., the risk that constitutes the tail of the distribution. Value at risk (VaR) has been widely adopted by financial institutions as a measure of financial exposure, and by regulators and rating agencies for determining the capital adequacy. Expected shortfall, another tail risk measure, is the average of losses exceeding some pre-described loss level.
Portfolio managers may also deal with an optimization problem that involves multiple return measures. Some portfolio managers may be concerned with accounting incomes as well as economic returns.
To incorporate multiple measures of risk and return, portfolio managers must consider a multi-objective optimization problem (rather than a two objective problem presented above). The general multi-objective optimization problem can be described by the following formulation:
MaximizeReturn measure(s); andMinimizeRisk measure(s);Subject toPortfolio constraints.
Adding complexity to the problem, the risk measures are typically non-linear, and sometimes non-convex. In a problem with all linear objective functions, one can use a linear programming solver to efficiently obtain optimal solutions. If one or more of the objective functions is non-linear, a non-linear solver is required. For a problem with high dimensionality, but wherein the risk measure is non-linear but convex, a special technique may be utilized to reduce computational run time. In a prior application, U.S. application Ser. No. 10/390,689 filed Mar. 19, 2003, which is incorporated herein by reference in its entirety, we described a method for solving such type of problem. The method describes providing a mathematical model of a relaxation of a problem; generating a sequence of additional constraints; and sequentially applying respective nonlinear risk functions to generate respective adjusted maximum return solutions to obtain an efficient frontier.
In modern-day portfolio management problems, measuring and incorporating tail risk introduces more complexity. Some of them are nonlinear and non-convex. Some measures are even not in analytical forms. The needs of present portfolio analysis calls for special techniques to solve this class of problems.
In accordance with another aspect of the invention, evolutionary algorithms will now be described. In one approach, Evolutionary Algorithms (EAs) include techniques based on a general paradigm of simulated natural evolution (Bäck 1996, Goldberg 1989). EAs perform their search by maintaining at any time t a population P(t)={P1(t), P2(t), . . . , Pp(t)} of individuals. “Genetic” operators that model simplified rules of biological evolution are applied to create the new and desirably more superior (optimal) population P(t+1). This process continues until a sufficiently good population is achieved, or some other termination condition is satisfied. Each Pi(t)(t), represents via an internal data structure, a potential solution to the original problem. The choice of an appropriate data structure for representing solutions is very much an “art” than “science” due to the plurality of data structures suitable for a given problem.
However, the choice of an appropriate representation is an important step in a successful application of EAs, and effort is required to select a data structure that is compact and can avoid creation of infeasible individuals. Closely linked to choice of representation of solutions, is choice of a fitness function ψ: P(t)→R. The fitness function assigns credit to candidate solutions. Individuals in a population are assigned fitness values according to some evaluation criterion. Fitness values measure how well individuals represent solutions to the problem. Highly fit individuals are more likely to create offspring by “recombination” or “mutation” operations, described below. Weak individuals are less likely to be picked for reproduction, and so they eventually die out. A mutation operator introduces genetic variations in the population by randomly modifying some of the building blocks of individuals.
Evolutionary algorithms are essentially parallel by design, and at each evolutionary step a breadth search of increasingly optimal sub-regions of the search space is performed. Evolutionary search is a powerful technique of solving problems, and is applicable to a wide variety of practical problems that are nearly intractable with other conventional optimization techniques. Practical evolutionary search schemes do not guarantee convergence to the global optimum in a predetermined finite time, but they are often capable of finding very good and consistent approximate solutions. However, they are shown to asymptotically converge under mild conditions (Subbu and Sanderson 2000).
Evolutionary algorithms have received a lot of attention for use in a single objective optimization and learning applications, and have been applied to various practical problems. In recent years, the relatively new area of evolutionary multi-objective optimization has grown considerably. Since evolutionary algorithms inherently work with a population of solutions, they are naturally suited for extension into the multi-objective optimization problem domain, which requires the search for and maintenance of multiple solutions during the search. This characteristic allows finding an entire set of Pareto optimal solutions in a single execution of the algorithm. Additionally, evolutionary algorithms are less sensitive to the shape or continuity of the Pareto front than traditional mathematical programming-based techniques.
In the past decade, the field of evolutionary multi-objective decision-making has been significantly energized, due in part to the multitude of immediate real-life applications in academia and industry. Several researchers have proposed several evolutionary multi-objective optimization techniques, as is reviewed and summarized in Coello et al. 2002.
However, the known techniques as discussed above fail to effectively and efficiently provide optimization processing to the extent possible. Further, techniques related to optimization processing fail to provide the tools needed to effectively work with the portfolios. The various systems and methods of the invention provide novel approaches to optimize portfolios.